Miscellanea 
431 
Again let me cite Dr Pearl: "The )(- test leads to this absurdity: if I perform a MeiideHan 
experiment in which I got ten thousand million offspring agreeing perfeclli/ with expectation save 
for one lone individual (perhaps a mutation, perhaps a mistake in the record, or what not) which 
is of a sort not expected, then Pearson and the test agree that the probability is infinitely 
great that the ten thousand million do not follow Mendelian lawl*" 
Now this paragraph is I fear "characteristically" partizan, for in my criticism of Dr Pearl's 
paper I said nothing whatever about ten thousand millions obejdng or not obeying the Mendelian 
law. Dr Pearl in his paper on the inheritance of fertility in hens got very frequently (from my 
standpoint) exceptions to his Mendelian theory. He drew in fact blue balls from a bag which 
his theory asserted contained only green or red balls. As a result he rejected the "goodness 
of fit" theory, which provided as it ought to do an infinite improbability. I asserted that he must 
either "remould his theory or explain away his observations f " ; it was not a question of my test 
being fallacious. That he appears to be doing now when he states that exceptions may be 
"mutations" or "mistakes in records" or "what nots," or as he did in his original paper when 
he remarked of such exceptions that they are of "a type which is continually arising in Mendelian 
work." In other words he then emphasised and now again emphasises the "elasticity" of his 
observations. That is certainly the horn of the dilemma upon which I was prepared to drive 
him. But he is not content with such elasticity of Mendelism, or with badness of records or 
with "what nots," he seeks to show that a true test of Mendelian theory will be as plastic as 
the observations are elastic, in which case he will have successfully polled both horns. 
In order to achieve this end he starts with a paper of mine published in the Philosophical 
Magazine for 1907, which he describes as "a very important paper, which is apparently almost 
entirely unknown to biologists." While feehng grateful for the compliment, I regret that it 
should be brought to the notice of biologists in regard to a matter which it cannot possibly cover. 
Let me briefly state the purport of that paper. A bag, a population, or a theoretical frequency 
is presented, the distribution law of which is absoliUehj unknoivn. The number in this funda- 
mental distribution is supposed indefinitely great or else individuals must be returned before 
each draw. A first sample of m is drawn of which p present one characteristic and q do not. 
A second sample of m is now drawn, what is the chance that r individuals of this m will present 
the characteristic? 
I showed that the mean of second samples would be 
p m q-p ,.v 
+ s i — - 1 , 
n n + 2 Ii 
and the standard deviation tr would be given by 
n [n + 2)J \n n (n + 2)/ \ n + 'i , 
the whole frequency distribution being provided by a hypergeometrical series. It is this hyper- 
geometrical series which Dr Pearl proposes to employ as a test of "goodness of fit" in Mendelian 
class frequencies. 
At first sight it seems almost impossible to determine how it can bs applied, for the Men- 
dehan Law which is supposed known is the theoretical frequency, the bag or original population, 
which my memoir supposes to bo iinknowii except as suggested by the first sample. The only 
adequate method of applying the theory — and then it would be using a sledge hammer to crack 
nuts — would be to suppose the first sample indefinitely large. In this case p/ii represents the 
class frequency of Mendelian theory p, and the ratio of the observed sample vi to the total 
possible Mendelian population n is vanishingly small, and accordingly there results 
mean = mp, a - = nvpq, 
and the hypergeometrical series collapses into N (p + q)'"-. Thus we fall back on the obvious 
binomial test with its infinite improbability for the cases oi p = 0, which Dr Pearl finds not 
* Loc. cit., p. 145. t Biometrika, Vol. ix. p. 312. 
